We consider the following problem: Let P be a monic polynomial of degree n with complex coefficients. What can be the maximum ‘size’ of a monic divisor Q of P? Here the size of a polynomial R is the maximum ||R|| of the moduli of its values on the unit circle. In 1991, B. Beauzamy proved that there exists a divisor Q with ||Q|| ≧ e∈n−1, ∈ = 0.0019, when all the roots of P belong to the unit circle. Using a very recent result of D. Boyd, we obtain a general result which, in the same case, gives ||Q||≧βn; here β = 1.38135 … is optimal.

We study the development of concentration profiles in a semi-infinite slab of semiconductor material after impurities have been implanted uniformly through the slab, under the assumption that, at the face of the slab, no impurities can pass and the vacancy concentration is kept at its equilibrium value. It is shown that profiles of self-similar form exist, and their qualitative shape, as well as their asymptotic properties far from the face of the slab, are determined.

The purpose of this paper is to describe various subspaces that are closely related to the absolutely continuous subspace of a Floquet operator. This paper generalises and extends several known results.

We consider an emulsion of two Stokes fluids, one of which is periodically distributed in the form of small spherical bubbles. The effects of surface tension on the bubble boundaries are modelled mathematically, as in the work of G. I. Taylor, by a jump only in the normal component of the traction. For a given volume fraction of bubbles, we consider the two-scale convergence, and in the fine phase limit we find that the bulk flow is described by an anisotropic Stokes fluid. The effective viscosity tensor is consistent with the bulk stress formula obtained by Batchelor [2].

In this paper a general linear model for vibrating networks of one-dimensional elements is derived. This is applied to various situations including nonplanar networks of beams modelled by a three-dimensional variant on the Timoshenko beam, described for the first time in this paper. The existence and regularity of solutions is established for all the networks under consideration. The methods of first-order hyperbolic systems are used to obtain estimates from which exact controllability follows for networks containing no closed loops.

We consider the bifurcation of positive solutions of the two-point boundary value problem

where λ> 0 is a real bifurcation parameter, and f ∊ C2 satisfies (fl) f(0) < 0, (f2) f′(s) > 0 for s > 0, (f3) f″(s) < 0 for s > 0 and (f4) limS→+∞f(s) = M where 0 < M ≦+∞. This problem has been studied by Casto and Shivaji under two additional hypotheses (f5) lims→+∞sf′(s) = 0, and (f6)f(θ)/θ < f′(θ), where θ is a positive number satisfying Assuming (fl)−(f6), Castro and Shivaji obtain some existence and nonexistence results and hence partial information on the bifurcation diagram, and they conjecture that this problem has at most two positive solutions. We prove this conjecture. Furthermore, we are able to generalise and improve their results under hypotheses (fl)−(f4). As a corollary, we show that there exists μ1 > 0 such that there exist no positive solutions for 0 <λ <μ1 and at most two positive solutions for μ1≦λ< + ∞, which improves a result of Brunovsky and Chow.

We study the 1-relator relative presentation 〈H, x|xaxbx−1c〉 where H is a group, a, b, c ∈ H, x ∉ H and b, c ≠ 1. We give necessary and sufficient conditions for this presentation to be aspherical apart from two outstanding special cases which remain open.

In this paper we study the asymptotic behaviour of reaction–diffusion systems with a small parameter by using the n-dimensional Feynman–Kac formula and large deviation theory. The generalised solutions are introduced in Section 2. We obtain the travelling wave joining an unstable steady state and an asymptotically stable steady state of a diffusionless dynamical system in a reaction–diffusion system with nonlinear ergodic interactions, and a special case with nonlinear reducible interactions.

An integral representation result is given for the lower semicontinuous envelope of the functional ʃΩf(∇ku) dx on the space BVk(Ω:ℝm) of the integrable functions, whose the f-th derivative in the sense of distributions is a Radon measure with bounded total variation.

In this paper we characterise the linear transformations of an infinite-dimensional vector space that can be written as the product of nilpotent transformations. This and a linear version of Malcev's congruence on transformation semigroups are then used to construct a new class of congruence-free semigroups.

This paper studies the surface of constant mean curvature on a semi-infinite strip, and shows by means of a first-order differential inequality that the solution in a given measure either becomes asymptotically unbounded at least to polynomial order, or decays at most exponentially to the solution of an associated one-dimensional problem. A proof is also presented for uniqueness in the class of functions having bounded gradient and subject to specified growth conditions for large values of the longitudinal distance. Extensions of these results to the whole strip and to more general types of equations are also described.

We consider the Cauchy problem for the quasilinear heat equation

where σ > 0 is a fixed constant, with the critical exponent in the source term β = βc = σ + 1 + 2/N. It is well-known that if β ∈(1,βc) then any non-negative weak solution u(x, t)≢0 blows up in a finite time. For the semilinear heat equation (HE) with σ = 0, the above result was proved by H. Fujita [4].

In the present paper we prove that u ≢ 0 blows up in the critical case β = σ + 1 + 2/N with σ > 0. A similar result is valid for the equation with gradient-dependent diffusivity

with σ > 0, and the critical exponent β = σ + 1 + (σ + 2)/N.

Let (X, ∑, μ) denote a σ-finite measure space. We show that the kernel condition on a weighted composition operator acting on L2(X, ∑, μ), which is necessary for hyponormality of the adjoint, implies that a certain subset of X has the localising property defined by Lambert. For operators satisfying this condition, we find a reducing subspace whose orthocomplement in L2 is annihilated by both the operator and its adjoint, allowing us to obtain characterisations of seminormality for the operator by looking only at the restriction to the reducing subspace. This simplifies the analysis significantly, giving transparent characterisations for the hyponormality and quasinormality of the adjoint, as well as a characterisation of normality for the operator which does not require the computation of any conditional expectations. Several examples are given. We then characterise the semi-hyponormal class for both the operator and its adjoint.

Although all the coefficients in the equation of a plane algebraic curve may be real numbers, it by no means follows that the equations of all its bitangents are real. But Plücker perceived that this could happen for the 28 bitangents of a nonsingular plane quartic. Where can this be observed in a body of 28 explicit linear equations? This modest note affords an example.

The blow-up behaviour of radially symmetric classical solutions to the quasilinear parabolic equation

is analysed assuming k(u) and Q(u) are small perturbations of {k, Q} ≡ {1, up},p > 1. Moreover, it is proved that the asymptotic behaviour near blowup of solutions to the semilinear equation ut = Δu + up, and in particular the final-time profile, is stable with respect to small quasilinear perturbations of the elliptic operator.

We consider an integrable case of the Henon-Heiles system and use an isomorphism with the two-gap KdV-flow to construct families of real elliptic trajectories which are associated with two-gap elliptic solitons of the KdV equation. Some of these solutions exhibit blow-up in finite time.

This paper is devoted to the study of integral functional denned on the space SBV(Ω ℝk) of vector-valued special functions with bounded variation on the open set Ω⊂ℝn, of the form

We suppose only that f is finite at one point, and that g is positively 1-homogeneous and locally bounded on the sets ℝk⊗vm, where {v1,…, vR} ⊂ Sn−1 is a basis of ℝn. We prove that the lower semicontinuous envelope of F in the L1(Ω;ℝk)-topology is finite and with linear growth on the whole BV(Ω;ℝk), and that it admits the integral representation

A formula for ϕ is given, which takes into account the interaction between the bulk energy density f and the surface energy density g.

We define a family of Cesàro operators , Reα≧0, and consider the question of their boundedness on Hp spaces. We also consider discrete versions of these operators acting on sequence spaces.

We study the existence of changing sign solutions of an elliptic semilinear boundary value problem, which arises as a limiting equation of the two species Lotka–Volterra competing equations system. Using variational methods and a result of D'Aujourd'hui, we find conditions which are both sufficient and necessary for this existence problem.

We give an existence result and some qualitative remarks about the optimisation of the torsional rigidity of a beam.